Finite Group Databasebeta

← #24992666

Subgroups — #24992666

Order 11008.

622 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19137213C211yesmaximal, characteristic, complemented
2#19133163C212yesmaximal
3C2×#14582625C211yesmaximal, characteristic, complemented
4#19132462C212yesmaximal, complemented
5C43×#2618C211yesmaximal, characteristic
6C2×#13063284—21no—
7C2×C22⋊(C43⋊C8)C2212yes—
8#14582798—24no—
9#14582759—21no—
10#14582759C2212yes—
11#14582633—24no—
12#14582625—22no—
13C2×#123×C43—21no—
14C2×#121×C43C2211yescharacteristic
15C22×C4⋊C4×C43C2211yescharacteristic
16C43×#398C412yescomplemented
17C43×#396—22no—
18C43×#394C412yescomplemented
19C43×#386—21no—
20C43×#380C2211yescharacteristic
21C22×C43⋊C8—22no—
22#13063284—42no—
23#13063284—42no—
24C2×C4⋊Dic43—22no—
25C2×C4⋊Dic43—22no—
26C2×C4⋊Dic43—22no—
27C2×C22⋊Dic43—22no—
28C2×C4×Dic43—22no—
29C22⋊(C43⋊C8)—22no—
30C22⋊(C43⋊C8)—22no—
31C43⋊#95—22no—
32C43⋊#95D414yes—
33C22×D4×C43—21no—
34C22×D4×C43C2×C411yescharacteristic
35C23×C4×C43D411yescharacteristic
36C23×C4×C43D411yescharacteristic
37C23×C4×C43C2311yescharacteristic
38#123×C43—42no—
39#123×C43—42no—
40#121×C43—22no—
41#121×C43—22no—
42C2×C4⋊C4×C43—22no—
43C2×C4⋊C4×C43—22no—
44C2×C4⋊C4×C43—22no—
45C2×C4⋊C4×C43—21no—
46C2×C4⋊C4×C43C2×C411yescharacteristic
47C2×C4⋊C4×C43C2×C412yes—
48C2×C4⋊C4×C43D412yes—
49C2×C22⋊C4×C43—21no—
50C2×C22⋊C4×C43—22no—
51C2×C22⋊C4×C43—21no—
52C2×C22⋊C4×C43—22no—
53#95×C43—22no—
54#95×C43C2×C412yes—
55C24×C43—21no—
56C24×C43C2×D411yescharacteristic
57C2×D4×C43—22no—
58C2×D4×C43—41no—
59C2×D4×C43—41no—
60C2×D4×C43—41no—
61C2×D4×C43—41no—
62C2×D4×C43—41no—
63C2×D4×C43—41no—
64C2×D4×C43—81no—
65C22×C4×C43—21no—
66C22×C4×C43—22no—
67C22×C4×C43—21no—
68C22×C4×C43—21no—
69C22×C4×C43—21no—
70C22×C4×C43—21no—
71C22×C4×C43—21no—
72C22×C4×C43—21no—
73C22×C4×C43—81no—
74C22×C4×C43—81no—
75C22×C4×C43C22⋊C412yes—
76C22×C4×C43C22⋊C412yes—
77C22×C4×C43D812yes—
78C22×C4×C43SD1612yes—
79C22×C4×C43C22×C411yescharacteristic
80C22×C4×C43C2×D411yescharacteristic
81C22×C4×C43C2×D412yes—
82C22×C4×C43#4112yes—
83C22×Dic43—24no—
84C4⋊C4×C43—24no—
85C4⋊C4×C43—24no—
86C4⋊C4×C43—22no—
87C4⋊C4×C43—44no—
88C4⋊C4×C43—81no—
89C22⋊C4×C43—44no—
90C22⋊C4×C43—42no—
91C22⋊C4×C43—41no—
92C22⋊C4×C43—41no—
93C22⋊C4×C43—82no—
94C22⋊Dic43—42no—
95C22⋊Dic43—42no—
96C4⋊Dic43—44no—
97C4⋊Dic43—82no—
98C4⋊Dic43—82no—
99C4×Dic43—44no—
100C2×C43⋊C8—42no—
101C2×C43⋊C8—82no—
102C23×C43—41no—
103C23×C43—41no—
104C23×C43—42no—
105C23×C43—42no—
106C23×C43—41no—
107C23×C43—41no—
108C23×C43—41no—
109C23×C43C2×C22⋊C411yescharacteristic
110C23×C43#12011yescharacteristic
111C23×C43#12311yescharacteristic
112C23×C43C2×D811yescharacteristic
113C23×C43C2×SD1611yescharacteristic
114C23×C43AGL(1,ℤ/8)11yescharacteristic
115C23×C43#13711yescharacteristic
116D4×C43—82no—
117D4×C43—81no—
118D4×C43—81no—
119D4×C43—81no—
120D4×C43—81no—
121C2×C4×C43—22no—
122C2×C4×C43—22no—
123C2×C4×C43—22no—
124C2×C4×C43—22no—
125C2×C4×C43—22no—
126C2×C4×C43—22no—
127C2×C4×C43—22no—
128C2×C4×C43—42no—
129C2×C4×C43—42no—
130C2×C4×C43—41no—
131C2×C4×C43—41no—
132C2×C4×C43—42no—
133C2×C4×C43—42no—
134C2×C4×C43—42no—
135C2×C4×C43—42no—
136C2×C4×C43—41no—
137C2×C4×C43—42no—
138C2×C4×C43—41no—
139C2×C4×C43—41no—
140C2×C4×C43—81no—
141C2×C4×C43#10214yes—
142C2×C4×C43C4×D412yes—
143C2×C4×C43#12112yes—
144C2×Dic43—44no—
145C2×Dic43—44no—
146C2×Dic43—44no—
147C43⋊C8—82no—
148#9204—431nomaximal, Sylow
149C22×C43—21no—
150C22×C43—21no—
151C22×C43—21no—
152C22×C43—21no—
153C22×C43—21no—
154C22×C43—21no—
155C22×C43—41no—
156C22×C43—41no—
157C22×C43—41no—
158C22×C43—41no—
159C22×C43—82no—
160C22×C43—82no—
161C22×C43—81no—
162C22×C43—81no—
163C22×C43—81no—
164C22×C43#38611yescharacteristic
165C22×C43C2×#10211yescharacteristic
166C22×C43#41711yescharacteristic
167C22×C43#44711yescharacteristic
168C22×C43#44811yescharacteristic
169C22×C43#48011yescharacteristic
170C22×C43#48111yescharacteristic
171C4×C43—24no—
172C4×C43—22no—
173C4×C43—22no—
174C4×C43—42no—
175C4×C43—82no—
176C4×C43—81no—
177Dic43—84no—
178#2618D4311yescharacteristic
179#2316—431no—
180#1519—432no—
181C2×#324—431no—
182#1289—432no—
183C2×C43—21no—
184C2×C43—21no—
185C2×C43—21no—
186C2×C43—21no—
187C2×C43—21no—
188C2×C43—21no—
189C2×C43—161no—
190C2×C43#187411yescharacteristic
191C2×C43#191211yescharacteristic
192C2×C43#191311yescharacteristic
193C2×#123—21no—
194C2×#121C2×D4311yescharacteristic
195C2×#117—861no—
196C22×C4⋊C4C2×D4311yescharacteristic
197C2×C22⋊C8—432no—
198#398Dic4312yescomplemented
199#396—22no—
200#394Dic4312yescomplemented
201#389—864no—
202#386—21no—
203#380—432no—
204#380—861no—
205#380C2×D4311yescharacteristic
206#327—864no—
207#324—862no—
208C43#920411yescharacteristic, complemented, Sylow
209C22×D4—21no—
210C22×D4C2×Dic4311yescharacteristic
211C23×C4C22⋊D4311yescharacteristic
212C23×C4C22⋊D4311yescharacteristic
213C23×C4C22×D4311yescharacteristic
214C22×C8—862no—
215#123—42no—
216#123—42no—
217#121—22no—
218#121—22no—
219#117—1722no—
220#117—1722no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—862no—
226C2×C4⋊C4—862no—
227C2×C4⋊C4—862no—
228C2×C4⋊C4C2×Dic4311yescharacteristic
229C2×C4⋊C4C2×Dic4312yes—
230C2×C4⋊C4C22⋊D4312yes—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—862no—
236C2×C42—862no—
237C22⋊C8—862no—
238C22⋊C8—862no—
239#95—22no—
240#95—434no—
241#95—862no—
242#95C2×Dic4312yes—
243C24—21no—
244C24C2×C22⋊D4311yescharacteristic
245C2×D4—22no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—81no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4—864no—
264C22×C4#1062188212yes—
265C22×C4#1062188312yes—
266C22×C4C22⋊Dic4312yes—
267C22×C4C22⋊Dic4312yes—
268C22×C4D4×D4312yes—
269C22×C4#1062190012yes—
270C22×C4C22×Dic4311yescharacteristic
271C22×C4C2×C22⋊D4311yescharacteristic
272C2×C8—3442no—
273C2×C8—1722no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—22no—
277C4⋊C4—44no—
278C4⋊C4—81no—
279C4⋊C4—3442no—
280C4⋊C4—3442no—
281C4⋊C4—1724no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—44no—
285C22⋊C4—42no—
286C22⋊C4—82no—
287C22⋊C4—1722no—
288C22⋊C4—1722no—
289C42—1724no—
290C23—41no—
291C23—41no—
292C23—42no—
293C23—42no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23C2×C22⋊Dic4311yescharacteristic
298C23#1306329311yescharacteristic
299C23#1306330611yescharacteristic
300C23C2×#1062188211yescharacteristic
301C23C2×#1062188311yescharacteristic
302C23#1306335011yescharacteristic
303C23#1306335511yescharacteristic
304D4—82no—
305D4—81no—
306D4—81no—
307D4—81no—
308D4—81no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—81no—
329C2×C4—1724no—
330C2×C4—1724no—
331C2×C4—1724no—
332C2×C4#1306324614yes—
333C2×C4D4×Dic4312yes—
334C2×C4#1306329712yes—
335C8—3442no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—82no—
347C22—82no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22#1458277911yescharacteristic
352C22C2×#1306324611yescharacteristic
353C22#1458290011yescharacteristic
354C22#1458303211yescharacteristic
355C22#1458303711yescharacteristic
356C22#1458321011yescharacteristic
357C22#1458321511yescharacteristic
358C4—22no—
359C4—22no—
360C4—24no—
361C4—42no—
362C4—82no—
363C4—81no—
364C4—3444no—
365C2—21no—
366C2—21no—
367C2—21no—
368C2—21no—
369C2—21no—
370C2—21no—
371C2—161no—
372C2#1913491311yescharacteristic
373C2#1913513311yescharacteristic
374C2#1913513811yescharacteristic
subgroupG/H
centerC22#14582779
commutatorC22×C4×C43C22×C4
FrattiniC23×C4C22×D43
FittingC43×#2618C2
socleC22×C43#386
prime radicalC2×#121×C43C22
cosocleC23×C4×C43C23
hypercenter#2618D43
hypocenterC43#9204
LCSG ▷ C22×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2618

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #19137213C2semidirect
3. C2×#14582625C2semidirect
4. #19132462C2semidirect
16. C43×#398C4semidirect
18. C43×#394C4semidirect
198. #398Dic43semidirect
200. #394Dic43semidirect
208. C43#9204semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#191372132
#191331632
C2×#145826252
#191324622
C43×#26182
#920443